We introduce and study a family of online metric problems with long-term constraints. In these problems, an online player makes decisions $\mathbf{x}_t$ in a metric space $(X,d)$ to simultaneously minimize their hitting cost $f_t(\mathbf{x}_t)$ and switching cost as determined by the metric. Over the time horizon $T$, the player must satisfy a long-term demand constraint $\sum_{t} c(\mathbf{x}_t) \geq 1$, where $c(\mathbf{x}_t)$ denotes the fraction of demand satisfied at time $t$. Such problems can find a wide array of applications to online resource allocation in sustainable energy and computing systems. We devise optimal competitive and learning-augmented algorithms for specific instantiations of these problems, and further show that our proposed algorithms perform well in numerical experiments.

翻译：我们引入并研究了一类带有长期约束的在线度量问题。在这些问题中，在线决策者在度量空间$(X,d)$中做出决策$\mathbf{x}_t$，以同时最小化其打击成本$f_t(\mathbf{x}_t)$和由度量决定的切换成本。在时间范围$T$内，决策者必须满足长期需求约束$\sum_{t} c(\mathbf{x}_t) \geq 1$，其中$c(\mathbf{x}_t)$表示在时间$t$满足的需求比例。这类问题广泛应用于可持续能源和计算系统中的在线资源分配。我们针对这些问题的具体实例设计了最优的竞争性和学习增强算法，并进一步表明，我们提出的算法在数值实验中表现良好。